**Lucky Edge Labeling of H Graph, N Copies of H-Graph, Theta Graph, **
**Duplication of Theta Graphs and Path Union of Theta Graphs **

**Shalini Rajendra Babu. Dr. N. Ramya**
[email protected]

Bharath Institute of Higher Education and research, Selaiyur, Chennai-73

**Abstract: **

In this paper, it is proved that the H-graphs, n copies of H-graphs, Theta graph, Path union of Theta graph and Duplication of Theta graph are Lucky edge graphs.

Let G be a simple graph with vertex set edge set respectively. Vertex set and edge set by positive integer and denote the edge label such that it is the sum of labels of vertices incident with edge. The labeling is said to be lucky edge labeling, if the edge set is a proper coloring of G that is if we

have wherever are adjacent edges.

**Keywords: **

Lucky edge graphs, Lucky edge labeling, H-Graph and Theta Graphs.

**Introduction: **

In 1967, Rosa [6] introduced the concept of labeling.

Nellai Murugan [3,4], introduced the concept of Lucky edge labeling. A vertex labeling of a graph G is an assignment of labels to the vertices of G that includes for each edge uv a label depends on the vertex labels x and y.

In this paper we proved that the H graphs, n copies of H-graph, Path union of Helm, path union of closed helm, path union of Gear graph are lucky edge labeled graphs.

**Preliminaries: **

**Definition 1.1 **

**Definition 1.2 **

The H-graph of a path is the graph obtained from two copies of with vertices by joining the vertices and by an edge if n is odd and the vertices and if n is even.[2].

**Definition 1.3 **

A Theta graph is a block with two non-adjacent vertices of degree 3, and all other vertices of degree 2. [1,8]

**Definition 1.4 **

A vertex is said to be a duplication of if all the vertices which are adjacent to are now adjacent to . [8]

**Theorem:1 **

The Theta graph admits Lucky edge labeling whose Lucky number is 6.

**Proof: **

If are the vertices of the Theta Graph be the central vertex and rest of the vertices are the external vertices.

Edge set can be defined as,

{

Let us define the vertex labeling labeling has to be given by,

v vi

Define the map on E as follows,

Let such that,

**Illustration:1 **

**Figure: 1 **

Figure-1 shows Theta graph and its lucky number is 6.

**Theorem:2 **

The duplication of any vertex of degree 3, in the Theta graph is a Lucky edge labelled graph and its Lucky edge number is 8.

**Proof: **

Let be a graph obtained from after duplication vertex of the and duplication vertex of the vertex .

Let us define vertex labeling as follows Case(i)

Labeling of Duplication of we define , is the Duplication vertex of

Define the map on E as follows,

Let such that,

**Illustration:2 **

**Figure: 2 **

Figure-2 shows the Duplication of Theta graph and its lucky number is 8.

**Theorem:3 **

For every there exists a path. Path union of “m” copies of Theta graph is a Lucky edge labeled graph whose Lucky number is 6.

Consider “m” copies of Theta graphs Then

**Proof: **

Let the function there exists a Path union of Theta graph whose vertex labeling is defined by

,

Define the amp on E as follows,

Let such that,

**Illustration:3 **

**Figure: 3 **

Figure-3 shows the Path union of Theta graph and its lucky number is 6.

**Theorem:4 **

The graph graph is a lucky edge labelled graph having Lucky number is 4 and graph , whose Lucky number is 5.

**Proof: **

Let be an H graph, with vertex partition vertices in A makes left arm of H and B makes right arm of it.

The component A contains the edges , where i 1, ,

, when n is odd , when n is an even.

Then vertices and edges.

Lucky edge labeling of graph is divided into two cases.

**Case(i) **

When n is an odd
**Sub case (ii) **

When n=3, the vertex labeling „f‟ is constructed as follows,

Edge labeling must be given by

**Figure: 4 **

Figure-4 shows the graph and its lucky number is 4.

**Subcase (ii) **

As explained the vertex labeling of above, it can be extended , inductively,

Pendant varices of , can be labelled as 2.

Edge labeling can easily be completed by adding the labels of the extreme verities of the given edge.

**Illustration:5 **

**Figure: 5 **

Figure-5 shows the graph and its lucky number is 5. [when

**Subcase (iii) **

As Explained the vertex labeling of above, it can be extended, for inductively, pendant vertices of those can be labelled as 1.

Computing of Edge labeling is mentioned in the subcase (ii)
**Illustration: 6 **

**Figure: 6 **

Figure-6 shows the graph and its lucky number is 5.

**Subcase (iv) **

The vertex labeling of is explained in subcase (ii), from that it can be extended inductively.

The pendant vertices if can be labelled as 3. Edge labeling can be computed by adding the labels of the extreme vertices of the given edge, but adjacent edge labels should not be in same.

**Illustration:7 **

**Figure: 7 **

Figure-7 shows the graph and its lucky number is 5.

**Case(ii) **

When n is an even
**Subcase(i) **

When n =4, the vertex labeling f is constructed as follows

and

Edge labeling must be given by

**Illustration:8 **

**Figure: 8 **

Figure-8 shows the graph and its lucky number is 4.

**Subcase(ii) **

The vertex labeling of is explained in subcase (i) it can be extended, .

Edge labeling can be computed by adding the labels of the extreme vertices of the given edge.

**Illustration:9 **

**Figure: 9 **

Figure-11 shows the graph and its lucky number is 6.

**Sub case (iii) **

The vertex labeling of is explained above in the previous subcase, it can be

extended for inductively, pendant vertices of

can be labelled by 1.

Computing of Edge labeling is mentioned in the previous subcase.

**Illustration:10 **

**Figure: 10 **

Figure-10 shows the graph and its lucky number is 6.

**Subcase (iv) **

The vertex labeling of has explained in the previous case, In addition to that pendant vertices of can be labelled by 2.

Edge labeling can be computed by adding the labels of the extreme vertices of the given edge.Labeling of the adjacent edges should not be in same.

**Theorem:5 **

The path union of n copies of graph and graph is lucky edge labeled graph.[8].

**Proof: **

Let , are the vertices of H-graph. and are connected by an edge, where

Define then the vertex labeling as

Edge labeling must be given by

**Illustration:11 **

**Figure: 11 **

Figure-11 represents s the Path union 4-Copies of graph and its lucky number is 6

**Case (ii) **

n copies of H graph

Let are the vertices of graph.

and are connected by an edges for all k.

Also, and are connected by a path for all k.

Define then the vertex labeling as

Edge labeling must be given as

**Illustration:12 **

**Figure: 12 **

Figure-12 indicates s the Path union 4-Copies of graph and its lucky number is 6.

**Conclusion: **

In this paper, we shown that the H-graphs, n copies of H-graphs, Theta graph, Path union of Theta graph and Duplication of Theta graph are Lucky edge graphs.

**References: **

1.Jagadeswari.P Manimekalai. K and Ramanathan K „Cube Difference Labeling of Theta Graphs‟ Volume 4, Issue 5, May– 2019 International Journal of Innovative Science and Research Technology

2.Kannan.M, Vikram Prasad.R, Gopi.R “Even vertex odd Mean labeling of H-graph”. Int Journal of Mathematics Archive-8(8),2017.

3. A. Nellai Murugan and R. Maria Irudhaya Aspin Chitra „Lucky Edge Labeling of Pn, Cn and Corona of Pn, Cn‟ International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 8, August 2014, PP 710-718.

4.A. Nellai Murugan and R. Maria Irudhaya Aspin Chitra „Lucky Edge Labeling of Planar Grid Graph‟ International Journal of Modern Sciences and Engineering Technology (IJMSET), Volume 2, Issue 9, 2015, pp.1-8

5.A. Nellai Murugan and R. Maria Irudhaya Aspin Chitra „Lucky Edge Labeling of Triangular Graphs‟ International Journal of Mathematics Trends and Technology (IJMTT) – Volume 36 Number 2- August 2016

6.Rosa, A. “On certain valuations of the vertices of a graph”, Theory of

7.Shalini Rajendra Babu, Ramya N, Rangarajan K „On Lucky Edge Labeling of Splitting Graphs and Snake Graphs‟ International Journal of Innovative Technology and Exploring Engineering (IJITEE) Volume-8 Issue-5 March, 2019

8.A. Sugumaran and P. Vishnu Prakash „Prime Cordial Labeling for Theta Graph‟ Annals of Pure and Applied Mathematics Vol. 14, No. 3, 2017, 379-386